CN114221638A - Maximum cross-correlation entropy Kalman filtering method based on random weighting criterion - Google Patents

Maximum cross-correlation entropy Kalman filtering method based on random weighting criterion Download PDF

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CN114221638A
CN114221638A CN202111568689.4A CN202111568689A CN114221638A CN 114221638 A CN114221638 A CN 114221638A CN 202111568689 A CN202111568689 A CN 202111568689A CN 114221638 A CN114221638 A CN 114221638A
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赵雪华
兰曼
卫一卿
王冰冰
王素芳
秦玉琨
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Luoyang Institute of Science and Technology
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Abstract

The invention provides a maximum cross-correlation entropy Kalman filtering method based on a random weighting criterion, which comprises the following steps: constructing a linear system equation and a measurement equation, selecting proper kernel width and initializing a system state and covariance, updating the state and covariance according to the system equation by one-step prediction, re-initializing a state value at the initial iteration moment of a fixed point, deforming the system model according to the initial system and the measurement equation, calculating the deformed error and a kernel function of the error, obtaining two diagonal arrays by a random weighting criterion and the kernel function, correcting the one-step prediction covariance and the measurement error by the two diagonal arrays so as to correct a gain matrix, and estimating the posterior state and the covariance of the system, aiming at the problem of non-Gaussian double tail impulse noise of a linear model, the method can obtain better performance than Kalman filtering and maximum cross-correlation entropy Kalman filtering, can be widely applied to the condition that noise of a linear system is non-Gaussian, and improves the filtering estimation precision under the condition of non-Gaussian noise.

Description

Maximum cross-correlation entropy Kalman filtering method based on random weighting criterion
Technical Field
The invention belongs to the technical field of signal processing, and particularly relates to a maximum cross-correlation entropy Kalman filtering method based on a random weighting criterion.
Background
State estimation is an important issue in signal processing. The Kalman filtering method is one of important methods for solving the state estimation problem of a linear system under Gaussian noise, fully utilizes a state model and observation data of the system, and obtains the optimal estimation of the system by solving the optimization problem to minimize the error of the state estimation. However, noise is polluted due to system models or measurement, and the like, and a double-tail non-gaussian noise condition often occurs, which causes the accuracy of the kalman filtering algorithm to be reduced and even to be dispersed, so that a filtering algorithm for the non-gaussian noise is required.
For the case that the measurement noise is non-gaussian, the filtering algorithms that have appeared at present are: gaussian sum filtering, M-estimation filtering based on Huber technique, and t-filtering based on Student's method. However, using gaussian and filtering requires that the probability density distribution of the noise is known, which is difficult to achieve in engineering practice: the Huber technology can not degrade after the influence parameter gamma exceeds 1.345, so that the estimation performance is degraded; whereas student t-filtering can only be used for cases where the system noise covariance and the metrology noise covariance are small. Therefore, it is very important to provide a new kalman filtering method suitable for improving both the robustness and the anti-noise capability of the linear system. For this reason, researchers propose a new solution to the heavy-tailed non-gaussian noise, that is, a filtering method based on the maximum cross-correlation entropy, such as the maximum cross-correlation entropy kalman filtering. Unlike the conventional filtering method, the cross-correlation entropy includes not only second-order statistical information but also higher-order statistical information, so that a better estimation effect can be obtained.
Disclosure of Invention
Aiming at the defects of the prior art, the invention provides a maximum cross-correlation entropy Kalman filtering method based on a random weighting criterion, which is suitable for a linear system under the condition that noise is non-Gaussian, and improves the robustness and anti-noise capability of the system.
In order to achieve the above object, the present invention is achieved by the following means.
A maximum cross-correlation entropy Kalman filtering method based on a random weighting criterion comprises the following steps:
the method comprises the following steps: the linear system equation and the measurement equation are constructed as follows:
Figure BDA0003422785090000011
wherein k-1 denotes the k-1 th time, xk∈RnIs an n-dimensional system state vector, z, at time kk∈RmAn m-dimensional measurement vector at the kth moment; fk-1And HkRespectively a known transfer matrix and a measurement matrix, qk-1∈RnIs the n-dimensional system noise at the k-1 time, rk∈RmMeasuring noise for m-dimension at the kth moment; system noise obeys gaussian distribution qk-1～N(0,Qk-1) The measured noise is non-Gaussian and follows the Gaussian mixture distribution rk～λN(0,Rk,1)+(1-λ)N(0,Rk,2)，qk-1And rkSatisfied for uncorrelated process and measure Gaussian noise
Figure BDA0003422785090000021
Wherein E [. C]Representing a mathematical expectation, δkjIs a function of the sign of the kronecker,
Figure BDA0003422785090000022
representing a mixed noise vector rjThe transposed vector of (1);
step two: initialization, selecting a kernel width σ, and initializing system states
Figure BDA0003422785090000023
And covariance P (0|0), let k equal to 1;
step three: according to one step of the systemMeasuring equation, updating prior state
Figure BDA0003422785090000024
Sum covariance Pk|k-1；
Figure BDA0003422785090000025
Figure BDA0003422785090000026
Step four: reinitializing the state values at the fixed-point iteration time by setting t equal to 1 and
Figure BDA0003422785090000027
step five: carrying out system model deformation according to the initial system and the measurement equation, and calculating the error of the new model, thereby calculating the kernel function of the error;
firstly, a state equation and a measurement equation are reconstructed:
Figure BDA0003422785090000028
Figure BDA0003422785090000029
Figure BDA00034227850900000210
wherein: e [. C]Representing a mathematical expectation, Pk|k-1Is the state one-step prediction error covariance matrix, R, at time kkIs the measured noise covariance matrix at time k, BP(k | k-1) is for Pk|k-1The matrix obtained after Cholesky decomposition,
Figure BDA00034227850900000211
is BPTransposed matrix of (k | k-1), as in BR(k) Is RkThe matrix obtained after Cholesky decomposition,
Figure BDA00034227850900000212
is BR(k) Transposed matrix of (A), BkIs formed by BP(k | k-1) and BR(k) Forming a new diagonal matrix;
in the equation
Figure BDA0003422785090000031
Are simultaneously multiplied by
Figure BDA0003422785090000032
Obtaining:
Dk＝Wkxk+ek
wherein
Figure BDA0003422785090000033
Error vector ekThe ith column element is:
ek(i)＝di(k)-wi(k)xk(i)
wherein: di(k) Is DkThe ith element of (1), wi(k) Is a matrix WkElement of row i, xk(i) Herein denotes xkOf the ith state quantity, and DkIs a vector with dimension L being n + m;
step six: obtaining two diagonal arrays by a random weighting criterion and a kernel function;
due to the random weighting criterion, a new cost function is defined:
Figure BDA0003422785090000034
wherein
Figure BDA0003422785090000035
Gσ(. G) Gaussian kernel function:
Figure BDA0003422785090000036
taking:
Figure BDA0003422785090000037
x is thenk(i) The optimal solution of (2):
Figure BDA0003422785090000038
the matrixing form is:
Figure BDA0003422785090000039
wherein
Figure BDA00034227850900000310
And is
Figure BDA00034227850900000311
Figure BDA00034227850900000312
Two diagonal matrices are obtained
Figure BDA00034227850900000313
Step seven: two diagonal matrix
Figure BDA00034227850900000314
To correct the one-step prediction covariance
Figure BDA00034227850900000315
And measurement error covariance
Figure BDA00034227850900000316
Figure BDA0003422785090000041
Figure BDA0003422785090000042
Thereby modifying the gain matrix;
Figure BDA0003422785090000043
step eight: estimating the posterior state of system filtering
Figure BDA0003422785090000044
Sum covariance
Figure BDA0003422785090000045
If k +1 is equal to N, wherein N is a preset algorithm iteration number, stopping calculation; otherwise, the steps are continuously executed.
Compared with the prior art, the invention has the beneficial effects that:
according to the invention, the accuracy of the MCKF is improved by introducing a random weighting criterion into the MCKF; the random weighting theory is adopted in the cost function, so that the correlation entropy is increased, the maximum correlation entropy criterion (MCC) is met, and the estimation precision is improved; experiments prove that compared with the traditional KF algorithm and the MCKF algorithm, the state estimation precision and the state estimation effectiveness are improved.
Drawings
FIG. 1 is a flow chart of a RWMCKF-based method according to the present invention;
FIG. 2 is a state x1Probability density functions at KF, MCKF (σ ═ 2), and RWCKF;
FIG. 3 is a state x2Probability density functions at KF, MCKF (σ ═ 2), and RWCKF;
FIG. 4 is a state x1Probability density functions at MCKF (σ ═ 2) and RWCKF;
FIG. 5 is a state x2Probability density functions under MCKF (σ ═ 2) and RWCKF.
Detailed Description
The invention is described in further detail below with reference to the figures and examples. For a better understanding of the method of the present invention, the network structure of the present invention will be described in detail.
The invention relates to a maximum cross-correlation entropy Kalman filtering method based on a random weighting criterion, which comprises the following steps:
the method comprises the following steps: the linear system equation and the measurement equation are constructed as follows:
Figure BDA0003422785090000046
wherein k-1 denotes the k-1 th time, xk∈RnIs an n-dimensional system state vector, z, at time kk∈RmAn m-dimensional measurement vector at the kth moment; fk-1And HkRespectively a known transfer matrix and a measurement matrix, qk-1∈RnIs the n-dimensional system noise at the k-1 time, rk∈RmFor m-dimensional measurement noise at time k, assuming that the system noise follows a Gaussian distribution qk-1～N(0,Qk-1) The measured noise is non-Gaussian and follows the Gaussian mixture distribution rk～λN(0,Rk,1)+(1-λ)N(0,Rk,2)，qk-1And rkSatisfied for uncorrelated process and measure Gaussian noise
Figure BDA0003422785090000051
Wherein E [. C]Representing a mathematical expectation, δkjIs a function of the sign of the kronecker,
Figure BDA0003422785090000052
representing a mixed noise vector rjThe transposed vector of (1);
step two: initialization, an appropriate kernel width σ is empirically selected, andinitializing system states
Figure BDA0003422785090000053
And covariance P (0|0), let k equal to 1;
step three: updating the prior state according to a one-step predictive equation of the system
Figure BDA0003422785090000054
Sum covariance Pk|k-1；
Figure BDA0003422785090000055
Figure BDA0003422785090000056
Step four: reinitializing the state values at the fixed-point iteration time by setting t equal to 1 and
Figure BDA0003422785090000057
step five: carrying out system model deformation according to the initial system and the measurement equation, and calculating the error of the new model, thereby calculating the kernel function of the error;
firstly, a state equation and a measurement equation are reconstructed:
Figure BDA0003422785090000058
Figure BDA0003422785090000059
Figure BDA00034227850900000510
wherein: e [. C]Representing a mathematical expectation, Pk|k-1Is the state one-step prediction error covariance matrix, R, at time kkIs the measured noise covariance matrix at time k, BP(k | k-1) is for Pk|k-1The matrix obtained after Cholesky decomposition,
Figure BDA00034227850900000511
is BPThe transposed matrix of (k | k-1). Same principle BR(k) Is RkThe matrix obtained after Cholesky decomposition,
Figure BDA00034227850900000512
is BR(k) Transposed matrix of (A), BkIs formed by BP(k | k-1) and BR(k) Forming a new diagonal matrix;
in the equation
Figure BDA0003422785090000061
Are simultaneously multiplied by
Figure BDA0003422785090000062
Obtaining:
Dk＝Wkxk+ek
wherein
Figure BDA0003422785090000063
Error vector ekThe ith column element is:
ek(i)＝di(k)-wi(k)xk(i)
wherein d isi(k) Is DkThe ith element of (1), wi(k) Is a matrix WkElement of row i, xk(i) Herein denotes xkOf the ith state quantity, and DkIs a vector of dimension n + m.
Step six: obtaining two diagonal arrays by a random weighting criterion and a kernel function;
due to the random weighting criterion, a new cost function is defined:
Figure BDA0003422785090000064
wherein
Figure BDA0003422785090000065
Gσ(. G) Gaussian kernel function:
Figure BDA0003422785090000066
taking:
Figure BDA0003422785090000067
x is thenk(i) The optimal solution of (2):
Figure BDA0003422785090000068
the matrixing form is:
Figure BDA0003422785090000069
wherein
Figure BDA00034227850900000610
And is
Figure BDA00034227850900000611
Figure BDA00034227850900000612
Two diagonal matrices are obtained
Figure BDA00034227850900000613
Step seven: two diagonal matrix
Figure BDA00034227850900000614
To correct the one-step prediction covariance
Figure BDA00034227850900000615
And measurement error covariance
Figure BDA00034227850900000616
Figure BDA00034227850900000617
Figure BDA00034227850900000618
Thereby modifying the gain matrix;
Figure BDA00034227850900000619
step eight: the a posteriori state and covariance of the system filtering are estimated, specifically,
Figure BDA0003422785090000071
Figure BDA0003422785090000072
at this time, the estimation of the target state parameter can be completed
Figure BDA0003422785090000073
Sum state estimation error covariance matrix
Figure BDA0003422785090000074
Will be used for the estimation of the state parameters at the next moment.
The random weighting maximum cross-correlation entropy Kalman filtering (RWMCKF) algorithm is adopted to carry out state estimation on a linear system under the condition of non-Gaussian heavy tail noise, and the application of the random weighting criterion enhances the robustness of the system and improves the filtering precision. According to the invention, MATLAB simulation software is used for carrying out simulation experiments, and the RWMCKF algorithm is compared with the existing filtering algorithms KF and MCKF, so that the precision of state estimation and the effectiveness of estimation are greatly improved.
The invention is demonstrated below by means of specific examples.
Example one
Consider a general linear system model:
Figure BDA0003422785090000075
Figure BDA0003422785090000076
where θ is π/18 and the system noise is Gaussian noise qi(k-1) to N (0,2) (i ═ 1,2), and the measurement noise is non-gaussian noise r (k) to 0.9N (0,1) +0.1N (0, 100).
According to the maximum correlation entropy Kalman filtering method based on the random weighting criterion, the initial value of the state is taken
Figure BDA0003422785090000077
The error covariance matrix takes P (0|0) ═ diag (100). The kernel widths σ of the gaussian kernel function are set to 0.1, 0.5, 1,2, 3, 5, 8, 10, respectively, and compared with the KF algorithm and the MCKF (σ ═ 2), respectively, resulting in the two states x of fig. 2, 31、x2Probability density functions under different filtering algorithms. The simulation result of the maximum correlation entropy Kalman filtering method based on the random weighting criterion is shown in a curve RWMCKF, and compared with the traditional KF and the MCKF, the method has the following obvious performance advantages: the KF algorithm is biased under the condition of heavy-tailed measurement noise, the MCKF and the RWMCF are unbiased, but the unbiased performance of the RWCKF algorithm is more advantageous than that of the MCKF algorithm.
The effect of the kernel width σ on the gaussian kernel function on the maximum correlation entropy kalman filtering of the provided random weighting criterion of the present invention can be derived by fig. 3, 4: RWCKF has similarities to MCKF in that σ -2 is most effective.
Example two:
changing the simulation model into one-dimensional linear uniform accelerated motion, wherein the system model and the measurement model are as follows:
Figure BDA0003422785090000081
Figure BDA0003422785090000082
wherein Δ t is 0.1s, and both the system noise and the measurement noise are nonlinear gaussian mixture noise:
q1(k-1)～0.9N(0,0.01)+0.1N(0,1)
q2(k-1)～0.9N(0,0.01)+0.1N(0,1)
q3(k-1)～0.9N(0,0.01)+0.1N(0,1)
r(k)～0.8N(0,0.01)+0.2N(0,100)。
according to the maximum correlation entropy Kalman filtering method based on the random weighting criterion, a filtering algorithm is initialized firstly. The initial value of the state and the initial value of the covariance are set to x (0) [001 ]]TP (0|0) ═ diag (0.01,0.01, 0.01). With the kernel width σ of the gaussian kernel function set to 2 (i.e., the relatively optimal kernel width in example 1), a mean square error table under the KF, MCKF (σ ═ 2), and RWMCKF (σ ═ 2) algorithms can be obtained:
table i mean square error under the KF, MCKF (σ ═ 2) and RWMCKF (σ ═ 2) algorithms
Figure BDA0003422785090000083
Table i gives the mean square error under three algorithms KF, MCKF (σ ═ 2), and RWMCKF (σ ═ 2), respectively, as can be derived from table i: under the influence of the same mixed Gaussian system and measurement noise, the mean square error of the maximum correlation entropy Kalman filtering variance based on the random weighting criterion is minimum, which shows that the performance of the algorithm is greatly improved compared with KF and MCKF.
The above description is only for the specific embodiments of the present invention, but the scope of the present invention is not limited thereto, and any changes or substitutions that can be easily conceived by those skilled in the art within the technical scope of the present invention are included in the scope of the present invention. Therefore, the protection scope of the present invention shall be subject to the protection scope of the appended claims.

Claims (1)

1. A maximum cross-correlation entropy Kalman filtering method based on a random weighting criterion is characterized in that: the method comprises the following steps:
the method comprises the following steps: the linear system equation and the measurement equation are constructed as follows:
Figure FDA0003422785080000011
wherein k-1 denotes the k-1 th time, xk∈RnIs an n-dimensional system state vector, z, at time kk∈RmAn m-dimensional measurement vector at the kth moment; fk-1And HkRespectively a known transfer matrix and a measurement matrix, qk-1∈RnIs the n-dimensional system noise at the k-1 time, rk∈RmMeasuring noise for m-dimension at the kth moment; system noise obeys gaussian distribution qk-1～N(0,Qk-1) The measured noise is non-Gaussian and follows the Gaussian mixture distribution rk～λN(0,Rk,1)+(1-λ)N(0,Rk,2)，qk-1And rkSatisfied for uncorrelated process and measure Gaussian noise
Figure FDA0003422785080000012
Wherein E [. C]Representing a mathematical expectation, δkjIs a function of the sign of the kronecker,
Figure FDA0003422785080000013
representing a mixed noise vector rjThe transposed vector of (1);
step two: initializing, selecting a kernel width σ, and initializing the systemStatus of state
Figure FDA0003422785080000014
And covariance P (0|0), let k equal to 1;
step three: updating the prior state according to a one-step predictive equation of the system
Figure FDA0003422785080000015
Sum covariance Pk|k-1；
Figure FDA0003422785080000016
Figure FDA0003422785080000017
Step four: reinitializing the state values at the fixed-point iteration time by setting t equal to 1 and
Figure FDA0003422785080000018
step five: carrying out system model deformation according to the initial system and the measurement equation, and calculating the error of the new model, thereby calculating the kernel function of the error;
firstly, a state equation and a measurement equation are reconstructed:
Figure FDA0003422785080000019
Figure FDA0003422785080000021
Figure FDA0003422785080000022
wherein: e [. C]RepresentsMathematical expectation, Pk|k-1Is the state one-step prediction error covariance matrix, R, at time kkIs the measured noise covariance matrix at time k, BP(k | k-1) is for Pk|k-1The matrix obtained after Cholesky decomposition,
Figure FDA0003422785080000023
is BPTransposed matrix of (k | k-1), as in BR(k) Is RkThe matrix obtained after Cholesky decomposition,
Figure FDA0003422785080000024
is BR(k) Transposed matrix of (A), BkIs formed by BP(k | k-1) and BR(k) Forming a new diagonal matrix;
in the equation
Figure FDA0003422785080000025
Are simultaneously multiplied by
Figure FDA0003422785080000026
Obtaining:
Dk＝Wkxk+ek
wherein
Figure FDA0003422785080000027
Error vector ekThe ith column element is:
ek(i)＝di(k)-wi(k)xk(i)
wherein: di(k) Is DkThe ith element of (1), wi(k) Is a matrix WkElement of row i, xk(i) Herein denotes xkOf the ith state quantity, and DkIs a vector with dimension L being n + m;
step six: obtaining two diagonal arrays by a random weighting criterion and a kernel function;
due to the random weighting criterion, a new cost function is defined:
Figure FDA0003422785080000028
wherein
Figure FDA0003422785080000029
Gσ(. G) Gaussian kernel function:
Figure FDA00034227850800000210
taking:
Figure FDA00034227850800000211
x is thenk(i) The optimal solution of (2):
Figure FDA0003422785080000031
the matrixing form is:
Figure FDA0003422785080000032
wherein
Figure FDA0003422785080000033
And is
Figure FDA0003422785080000034
Figure FDA0003422785080000035
Two diagonal matrices are obtained
Figure FDA0003422785080000036
Step seven: two diagonal matrix
Figure FDA0003422785080000037
To correct the one-step prediction covariance
Figure FDA0003422785080000038
And measurement error covariance
Figure FDA0003422785080000039
Figure FDA00034227850800000310
Figure FDA00034227850800000311
Thereby modifying the gain matrix;
Figure FDA00034227850800000312
step eight: estimating the posterior state of system filtering
Figure FDA00034227850800000313
Sum covariance
Figure FDA00034227850800000314
If k +1 is equal to N, wherein N is a preset algorithm iteration number, stopping calculation; otherwise, the steps are continuously executed.
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Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN114614797A (en) * 2022-05-12 2022-06-10 之江实验室 Adaptive filtering method and system based on generalized maximum asymmetric correlation entropy criterion
CN117705108A (en) * 2023-12-12 2024-03-15 中铁第四勘察设计院集团有限公司 Filtering positioning method, system and filter based on maximum correlation entropy

Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN114614797A (en) * 2022-05-12 2022-06-10 之江实验室 Adaptive filtering method and system based on generalized maximum asymmetric correlation entropy criterion
CN117705108A (en) * 2023-12-12 2024-03-15 中铁第四勘察设计院集团有限公司 Filtering positioning method, system and filter based on maximum correlation entropy

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